Mathematical modelling of the HIV/AIDS epidemic and the effect of public health education

>Magister Scientiae - MScHIV/AIDS is nowadays considered as the greatest public health disaster of modern time. Its progression has challenged the global population for decades. Through mathematical modelling, researchers have studied different interventions on the HIV pandemic, such as treatment, education, condom use, etc. Our research focuses on different compartmental models with emphasis on the effect of public health education. From the point of view of statistics, it is well known how the public health educational programs contribute towards the reduction of the spread of HIV/AIDS epidemic. Many models have been studied towards understanding the dynamics of the HIV/AIDS epidemic. The impact of ARV treatment have been observed and analysed by many researchers. Our research studies and investigates a compartmental model of HIV with treatment and education campaign. We study the existence of equilibrium points and their stability. Original contributions of this dissertation are the modifications on the model of Cai et al. [1], which enables us to use optimal control theory to identify optimal roll-out of strategies to control the HIV/AIDS. Furthermore, we introduce randomness into the model and we study the almost sure exponential stability of the disease free equilibrium. The randomness is regarded as environmental perturbations in the system. Another contribution is the global stability analysis on the model of Nyabadza et al. in [3]. The stability thresholds are compared for the HIV/AIDS in the absence of any intervention to assess the possible community benefit of public health educational campaigns. We illustrate the results by way simulation The following papers form the basis of much of the content of this dissertation, [1 ] L. Cai, Xuezhi Li, Mini Ghosh, Boazhu Guo. Stability analysis of an HIV/AIDS epidemic model with treatment, 229 (2009) 313-323. [2 ] C.P. Bhunu, S. Mushayabasa, H. Kojouharov, J.M. Tchuenche. Mathematical Analysis of an HIV/AIDS Model: Impact of Educational Programs and Abstinence in Sub-Saharan Africa. J Math Model Algor 10 (2011),31-55. [3 ] F. Nyabadza, C. Chiyaka, Z. Mukandavire, S.D. Hove-Musekwa. Analysis of an HIV/AIDS model with public-health information campaigns and individual with-drawal. Journal of Biological Systems, 18, 2 (2010) 357-375. Through this dissertation the author has contributed to two manuscripts [4] and [5], which are currently under review towards publication in journals, [4 ] G. Abiodun, S. Maku Vyambwera, N. Marcus, K. Okosun, P. Witbooi. Control and sensitivity of an HIV model with public health education (under submission). [5 ] P.Witbooi, M. Nsuami, S. Maku Vyambwera. Stability of a stochastic model of HIV population dynamics (under submission)

Mathematical modeling of TB disease dynamics in a crowded population.

Philosophiae Doctor - PhDTuberculosis is a bacterial infection which is a major cause of death worldwide. TB is a curable disease, however the bacterium can become resistant to the first line treatment against the disease. This leads to a disease called drug resistant TB that is difficult and expensive to treat. It is well-known that TB disease thrives in communities in overcrowded environments with poor ventilation, weak nutrition, inadequate or inaccessible medical care, etc, such as in some prisons or some refugee camps. In particular, the World Health Organization discovered that a number of prisoners come from socio-economic disadvantaged population where the burden of TB disease may be already high and access to medical care may be limited. In this dissertation we propose compartmental models of systems of differential equations to describe the population dynamics of TB disease under conditions of crowding. Such models can be used to make quantitative projections of TB prevalence and to measure the effect of interventions. Indeed we apply these models to specific regions and for specific purposes. The models are more widely applicable, however in this dissertation we calibrate and apply the models to prison populations

A stochastic TB model for a crowded environment

We propose a stochastic compartmental model for the population dynamics of tuberculosis. The model is applicable to crowded environments such as for people in high density camps or in prisons. We start off with a known ordinary differential equation model, and we impose stochastic perturbation.We prove the existence and uniqueness of positive solutions of a stochastic model. We introduce an invariant generalizing thebasic reproductionnumber andprove the stabilityof thedisease-free equilibriumwhen it is below unity or slightly higher than unity and the perturbation is small. Ourmain theorem implies that the stochastic perturbation enhances stability of the disease-free equilibrium of the underlying deterministic model. Finally, we perform some simulations to illustrate the analytical findings and the utility of the model

Control and elimination in an SEIR model for the disease dynamics of Covid-19 with vaccination

COVID-19 has become a serious pandemic affecting many countries around the world since it was discovered in 2019. In this research, we present a compartmental model in ordinary differential equations for COVID-19 with vaccination, inflow of infected and a generalized contact rate. Existence of a unique global positive solution of the model is proved, followed by stability analysis of the equilibrium points. A control problem is presented, with vaccination as well as reduction of the contact rate by way of education, law enforcement or lockdown. In the last section, we use numerical simulations with data applicable to South Africa, for supporting our theoretical results. The model and application illustrate the interesting manner in which a diseased population can be perturbed from within itself

A model of population dynamics of TB in a prison system and application to South Africa

BACKGROUND: Tuberculosis (TB) continues to spread in South African prisons in particular, as prisons are over-capacitated and have poor ventilation. The awaiting trial detainees are not screened on admission and are at high risk of getting infected with TB. RESULTS: We propose a compartmental model to describe the population dynamics of TB disease in prisons. Our model considers the inflow of susceptible, exposed and TB infectives into the prison population. Removal of individuals out of the prison population can be either by death or by being released from prison, as compared to a general population in which removal is only by death. We describe conditions, including non-inflow of infectives into the prison, which will ensure that TB can be eradicated from the prison population. The model is calibrated for the South African prison system, by using data in existing literature. The model can be used to make quantitative projections of TB prevalence and to measure the effect of interventions. Illustrative simulations in this regard are presented. The model can be used for other prison populations too, if data is available to calculate the model parameters. CONCLUSIONS: Various simulations generated with our model serve to illustrate how it can be utilized in making future projections of the levels of prevalence of TB, and to quantify the effect of interventions such as screening, treatment or reduction of transmission parameter values through improved living conditions for inmates. This makes it particularly useful as there are various targets set by the World Health Organization and by governments, for reduction of TB prevalence and ultimately its eradication. Towards eradication of TB from a prison system, the theorem on global stability of the disease-free state is a useful indicator

A Two-Group Model of TB in a Crowded Environment

In this paper, we introduce a two-group epidemic model of the dynamics of tuberculosis in a prison system. The total population of inmates consists of two groups: The sentenced inmates and those inmates who are awaiting trial. The threshold parameter for local stability is computed and analysed. We also prove the global stability of the disease free equilibrium using a Lyapunov function. We apply the model to South African data on tuberculosis and we observe consistency between the model prediction and the data. Numerical results are presented to illustrate the analytical results